Chapter Twelve
Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.
McGraw-Hill/Irwin

LU 12-1 Compound Interest (Future Value) – The Big Picture
1.
2.
3.
Compare simple interest with compound interest.
Calculate the compound amount and interest manually and by table lookup.
Explain and compute the effective rate (APY).
LU 12-2 Present Value -- The Big Picture
1.
2.
3.
Compare present value (PV) with compound interest (FV).
Compute present value by table lookup.
Check the present value answer by compounding.
12-2

COMPOUND INTEREST
• So far we have discussed only simple interest, which is interest on the principal alone. Simple interest is either paid at the end of the loan period or deducted in advance.
• In this chapter we look at the power of compounding—interest paid on earned interest.
• Compounding involves the calculation of interest periodically over the life of the loan (or investment). After each calculation, the interest is added to the principal. Future calculations are on the adjusted principal (old principal plus interest). Compound interest, then, is the interest on the principal plus the interest of prior periods. Future value (FV), or the compound amount, is the final amount of the loan or investment at the end of the last period.
• Before you learn how to calculate compound interest and compare it to simple interest, you must understand the terms that follow.
12-3

Compounding –
Involves the calculation of interest periodically over the life of the loan or investment
Compound Interest –
The interest on the principal plus the interest of prior periods
Future Value (compound amount) –
The final amount of the loan or investment at the end of the last period
Present Value –
The value of a loan or investment today
12-4

Compounding Periods
Compounding Annually
Compounding Semiannually
Compounding Quarterly
Compounding Monthly
Compounding Daily
Interest Calculated
Once a year
Every 6 months
Every 3 months
Every month
Every day
• Compounding changes the interest rate.
• Rate for each period is annual interest rate divided by the number of times the interest is compounded per year.
12-5

Compounding goes from present value to future value
$5.00
$4.50
$4.00
$3.50
$3.00
$2.50
$2.00
$1.50
$1.00
$0.50
$0.00
Present value
$1.00
0
After 1 period,
$1 is worth
$1.08
$1.08
After 2 periods,
$1 is worth
$1.17
$1.1664
1 2
Number of periods
After 3 periods,
$1 is worth
$1.26
Future
Value
After 4 periods,
$1 is worth
$1.36
$1.2597
$1.3605
3 4
12-6

Manual Calculation
Interest
Beg. Bal
End of year
Year 1
$ 1.00
0.08
$ 0.08
1.00
$ 1.08
Year 2
$ 1.08
x .10
$ 0.09
1.08
$ 1.17
Year 3
$ 1.17
x .10
$ 0.09
1.17
$ 1.26
Year 4
$ 1.26
x .10
$ 0.10
1.26
$ 1.36
12-7

Number of periods ( N )
Number of years multiplied by the number of times the interest is compounded per year
Rate for each period ( R )
Annual interest rate divided by the number of times the interest is compounded per year
If you compounded $100 for 4 years at 8% annually, semiannually, or quarterly, what is N and R ?
Periods
Annually:
Semiannually:
Quarterly:
4 x 1 = 4
4 x 2 = 8
4 x 4 = 16
Annually:
Quarterly:
Rate
8% / 1 = 8%
Semiannually: 8% / 2 = 4%
8% / 4 = 2%
12-8

SIMPLE VERSUS COMPOUND INTEREST
Simple
Bill Smith deposited $80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill’s simple interest and maturity value?
Compounded
Bill Smith deposited $80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill’s interest and compounded amount?
I = P x R x T
I = $80 x .08 x 4
I = $25.60
MV = $80 + $25.60
MV = $105.60
Interest
Beg. bal
Year 1
$ 80.00
$ x .08
6.40
80.00
End of year $ 86.40
Year 2
$ 86.40
x .08
$ 6.91
86.40
$ 93.31
Year 3
$ 93.31
x .08
$ 7.46
93.31
$ 100.77
Interest: $108.83 -- $80.00 = $28.83
Year 4
$ 100.77
x .08
$ 8.06
100.77
$ 108.83
12-9

Step 1 . Find the periods: Years multiplied by number of times interest is compounded in 1 year.
Step 2 . Find the rate: Annual rate divided by number of times interest is compounded in 1 year.
Step 3 . Go down the period column of the table to the number desired; look across the row to find the rate. At the intersection is the table factor for the compound amount of
$1.
Step 4 . Multiply the table factor by the amount of the loan. This gives the compound amount.
12-10

13
14
15
10
11
8
9
12
6
7
4
5
Period
1
2
3
Future value of $1 at compound interest (Partial)
1% 1.50% 2%
1.0100
1.0150
1.0200
1.0201
1.0302
1.0404
1.0300
1.0457
1.0612
1.0406
1.0614
1.0824
1.0510
1.0773
1.1041
1.0615
1.0934
1.1262
1.0721
1.1098
1.1487
1.0829
1.1265
1.1717
1.0937
1.1434
1.1951
1.1046
1.1605
1.2190
1.1157
1.1780
1.2434
1.1260
1.1960
1.2682
1.1381
1.2135
1.2936
1.1495
1.2318
1.3195
1.1610
1.2502
1.3459
1.2668
1.3048
1.3439
1.3842
1.4258
1.4685
1.5126
1.5580
3%
1.0300
1.0609
1.0927
1.1255
1.1593
1.1941
1.2299
4% 5% 6%
1.0400
1.0500
1.0600
1.0816
1.1025
1.1236
1.1249
1.1576
1.1910
1.1699
1.2155
1.2625
1.2167
1.2763
1.3382
1.2653
1.3401
1.4185
1.3159
1.4071
1.5036
1.3686
1.4775
1.5938
1.4233
1.5513
1.6895
1.4802
1.6289
1.7908
1.5395
1.7103
1.8983
1.6010
1.7959
2.0122
1.6651
1.8856
2.1329
1.7317
1.9799
2.2609
1.8009
2.0789
2.3966
7% 8% 9%
1.0700
1.0800
1.0900
1.1449
1.1664
1.1881
1.2250
1.2597
1.2950
1.3108
1.3605
1.4116
1.4026
1.4693
1.5386
1.5007
1.5869
1.6771
1.6058
1.7138
1.8280
1.7182
1.8509
1.9926
1.8385
1.9990
2.1719
1.9672
2.1589
2.3674
2.1049
2.3316
2.5804
2.2522
2.5182
2.8127
2.4098
2.7196
3.0658
2.5785
2.9372
3.3417
2.7590
3.1722
3.6425
2.1436
2.3579
2.5937
2.8531
3.1384
3.4523
3.7975
4.1772
10%
1.1000
1.2100
1.3310
1.4641
1.6105
1.7716
1.9487
12-11

Pam Donahue deposits $8,000 in her savings account that pays 6% interest compounded quarterly. What will be the balance of her account at the end of 5 years?
Periods (N) = 4 x 5 = 20
Rate (R) = 6%/4 = 1.5%
Table Factor = 1.3469
Compounded Amount:
$8,000 x 1.3469 = $10,775.20
12-12

Nominal Rate (stated rate) –
The rate on which the bank calculates interest
Effective rate (APY) = Interest for 1 year
Principal
12-13

12-14

12-15

Interest on a 1% deposit compounded daily--360 day basis
10
15
20
25
30
7
8
9
Period 6.00%
1 1.0618
6.50%
1.0672
7.00%
1.0725
7.50%
1.0779
8.00%
1.0833
8.50%
1.0887
9.00%
1.0942
9.50%
1.0996
10.00%
1.1052
4
5
2
3
6
1.1275
1.1972
1.2712
1.3498
1.4333
1.1388
1.2153
1.2969
1.3840
1.4769
1.1503
1.2337
1.3231
1.4190
1.5219
1.1618
1.2523
1.3498
1.4549
1.5682
1.1735
1.2712
1.3771
1.4917
1.6160
1.1853
1.2904
1.4049
1.5295
1.6652
1.1972
1.3099
1.4333
1.5862
1.7159
1.2092
1.3297
1.4622
1.6079
1.7681
1.2214
1.3498
1.4917
1.6486
1.8220
1.5219
1.6160
1.7159
1.5761
1.6819
1.7949
1.6322
1.7506
1.8775
1.6904
1.8220
1.9639
1.7506
1.8963
2.0543
1.8129
1.9737
2.1488
1.8775
2.0543
2.2477
1.9443
2.1381
2.3511
2.0136
2.2253
2.4593
1.8220
1.9154
2.0136
2.1168
2.2253
2.3394
2.4593
2.5854
2.7179
2.4594
2.6509
2.8574
3.0799
3.3197
3.5782
3.8568
4.1571
4.4808
3.3198
3.6689
4.0546
4.4810
4.9522
5.4728
6.0482
6.6842
7.3870
4.4811
5.0777
5.7536
6.5195
7.3874
8.3708
9.4851
10.7477
12.1782
6.0487
7.0275
8.1645
9.4855
11.0202
12.8032
14.8747
17.2813
20.0772
12-16

Use Table 12.2 to calculate what $1,500 compounded daily for
5 years will grow to at 7%.
N = 5
R = 7%
Factor, 1.4190
$1,500 x 1.4190 = $2,128.50
12-17

Present value goes from the future value to the present value
Future Value
$1.20
$1.10
$1.00
$0.90
$0.80
$0.70
$0.60
$0.50
$0.40
$0.30
$0.20
$0.10
$0.00
Present value
$.7350
$.7938
$.8573
$.9259
$1.0000
0 1 2 3 4
Number of periods
12-18

Step 1 . Find the periods: Years multiplied by number of times interest is compounded in 1 year.
Step 2 . Find the rate: Annual rate divided by number of times interest is compounded in 1 year.
Step 3 . Go down the Period column of the table to the number desired; look across the row to find the rate. At the intersection of the two columns is the table factor for the compound value of $1.
Step 4 . Multiply the table factor by the future value. This is the present value.
12-19

8
9
6
7
10
11
Period
1
2
3
4
5
12
13
14
15
0.9420
0.9327
0.9235
0.9143
0.9053
0.8963
1%
0.9901
0.9803
0.9706
0.9610
0.9515
0.8874
0.8787
0.8700
0.8613
Present value of $1 at end period (partial)
0.8880
0.8706
0.8535
0.8368
0.8203
0.8043
2%
0.9804
0.9612
0.9423
0.9238
0.9057
0.7885
0.7730
0.7579
0.7430
0.9145
0.9010
0.8877
0.8746
0.8617
0.8489
1.50%
0.9852
0.9707
0.9563
0.9422
0.9283
0.8364
0.8240
0.8119
0.7999
0.7050
0.6651
0.6274
0.5919
0.5584
0.5268
6%
0.9434
0.8900
0.8396
0.7921
0.7473
0.4970
0.4688
0.4423
0.4173
0.7462
0.7107
0.6768
0.6446
0.6139
0.5847
5%
0.9524
0.9070
0.8638
0.8227
0.7835
0.5568
0.5303
0.5051
0.4810
0.7903
0.7599
0.7307
0.7026
0.6756
0.6496
4%
0.9615
0.9246
0.8890
0.8548
0.8219
0.6246
0.6006
0.5775
0.5553
0.8375
0.8131
0.7894
0.7664
0.7441
0.7224
3%
0.9709
0.9426
0.9151
0.8885
0.8626
0.7014
0.6810
0.6611
0.6419
0.6663
0.6227
0.5820
0.5439
0.5083
0.4751
7%
0.9346
0.8734
0.8163
0.7629
0.7130
0.4440
0.4150
0.3878
0.3624
0.6302
0.5835
0.5403
0.5002
0.4632
0.4289
8%
0.9259
0.8573
0.7938
0.7350
0.6806
0.3971
0.3677
0.3405
0.3152
0.5963
0.5470
0.5019
0.4604
0.4224
0.3875
9%
0.9174
0.8417
0.7722
0.7084
0.6499
0.3555
0.3262
0.2992
0.2745
0.5645
0.5132
0.4665
0.4241
0.3855
0.3505
10%
0.9091
0.8264
0.7513
0.6830
0.6209
0.3186
0.2897
0.2633
0.2394
12-20

COMPARING COMPOUND INTEREST (FV)
(TABLE 12.1) WITH
PRESENT VALUE (PV) (TABLE 12.3)
Compound value Table 12.1
Table Present Future
12.1 Value
1.3605 x $80 = $108.84
Value
(N = 4, R = 8%)
We know the present dollar amount and find what the dollar amount is worth in the future.
Present value Table 12.3
Table Future
12.3
Value
Present
Value
.7350 x $108.84 = $80.00
(N = 4, R = 8%)
We know the future dollar amount and find what the dollar amount is worth in the present.
12-21

Rene Weaver needs $20,000 for college in 4 years. She can earn 8% compounded quarterly at her bank. How much must Rene deposit at the beginning of the year to have $20,000 in 4 years?
Periods (N) = 4 x 4 = 16
Rate (R) = 8%/4 = 2%
Table Factor = .7284
Invest
Today
Compounded Amount:
$20,000 x .7284 = $14,568
12-22

PRACTICE QUIZ-1
1-Complete the following without a table (round each calculation to the nearest cent as needed):
12-23

PRACTICE QUIZ-1
2-Solve the previous problem by using compound value (FV) in Table 12.1.
3-Lionel Rodgers deposits $6,000 in Victory Bank, which pays 3% interest compounded semiannually. How much will Lionel have in his account at the end of 8 years?
4-Find the effective rate (APY) for the year: principal, $7,000; interest rate,
12%; and compounded quarterly.
5-Calculate by Table 12.2 what $1,500 compounded daily for 5 years will grow to at 7%.
For step by step solution watch the video for LU 12-1 ( Go to: McGraw-Hill’s Connect;
Assignment # 4; Question 1; Click the eBook & resources options drop down menu; scroll down to LU12-1 and click
12-24

PRACTICE QUIZ-2
Use the present value Table 12.3 to complete:
3-Bill Blum needs $20,000 6 years from today to attend V.P.R. Tech. How much must Bill put in the bank today (12% quarterly) to reach his goal?
4-Bob Fry wants to buy his grandson a Ford Taurus in 4 years. The cost of a car will be $24,000.
Assuming a bank rate of 8% compounded quarterly, how much must Bob put in the bank today?
For step by step solution watch the video for LU 12-1 ( Go to: McGraw-Hill’s Connect;
Assignment # 4; Question 5; Click the eBook & resources options drop down menu; scroll down to LU12-2 and click
12-25

Lynn Ally, owner of a local Subway shop, loaned $40,000 to Pete Hall to help him open a Subway franchise. Pete plans to repay Lynn at the end of 8 years with 6% interest compounded semiannually. How much will Lynn receive at the end of 8 years? LU 12-1(2)
Solution:
8 years x 2 = 16 periods
6%
2
= 3% $40,000 x 1.6047 = $64,188
12-26

Melvin Indecision has difficulty deciding whether to put his savings in Mystic Bank or Four Rivers Bank. Mystic offers 10% interest compounded semiannually. Four
Rivers offers 8% interest compounded quarterly. Melvin has $10,000 to invest. He expects to withdraw the money at the end of 4 years. Which bank gives Melvin the better deal? Check your answer. LU 12-1(3)
Solution:
Mystic
4 years x 2 = 8 periods
10%
2
= 5%
$10,000 x 1.4775 = $14,775
-- 10,000
$ 4,775
Four Rivers
4 years x 4 = 16 periods
8%
4
= 2%
$10,000 X 1.3728 = $13,728
-- 10,000
$ 3,728
12-27

Lee Holmes deposited $15,000 in a new savings account at 9% interest compounded semiannually. At the beginning of year 4, Lee deposits an additional $40,000 at 9% interest compounded semiannually. At the end of 6 years, what is the balance in Lee’s account? LU 12-1(2)
Solution:
3 years x 2 = 6 periods
9%
2 = 4.5%
$15,000 x 1.3023 = $19,534.50
+ 40,000.00
$ 59,534.50
$59,534.50 x 1.3023 = $77,531.78
12-28

Paul Havlik promised his grandson Jamie that he would give him $6,000 8 years from today for graduating from high school. Assume money is worth 6% interest compounded semiannually. What is the present value of this $6,000?
LU 12-2(2)
Solution:
8 years x 2 = 16 periods
6%
2
= 3%
$6,000 x .6232 = $3,739.20
12-29

DRILL PROBLEMS
12-30

12-31